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Import Data

Import the AVISO-data from the file 'Agulhas_AVISO.mat' stored in the folder 'Data'.

Data/Parameters for Dynamical System

Spatio-Temporal Domain of Dynamical System

Velocity Interpolation

In order to evaluate the velocity field at arbitrary locations and times, we must interpolate the discrete velocity data. The interpolation with respect to time is always linear. The interpolation with respect to space can be chosen to be "cubic" or "linear". In order to favour a smooth velocity field, we interpolate the velocity field in space using a cubic interpolant.

Finite-Time Lyapunov Exponent (FTLE)

Next, we compute the FTLE over the meshgrid over the given time-interval. We iterate over all initial conditions and first calculate the gradient of the flow map using an auxiliary grid. 'aux_grid' specifies the ratio between the auxiliary grid and the original meshgrid. This parameter is generally chosen to be between $ [\dfrac{1}{10}, \dfrac{1}{100}] $. Subsequently, we compute the Cauchy Green strain tensor. From the Cauchy-Green strain tensor we can then compute the FTLE. The iteration over the meshgrid is parallelized.

The FTLE is given by: \begin{equation} \mathrm{FTLE}_{t_0}^{t_N} = \dfrac{1}{t_N-t_0}\log(\sqrt{\lambda_{2}(C_{t_0}^{t_N}(\mathbf{x})}), \end{equation} with $ \lambda_{2}(C_{t_0}^{t_N}(\mathbf{x})) $ denoting the maximum eigenvalue associated to the Cauchy Green strain tensor over the time-interval $ [t_0, t_N] $